We study a version of the Diamond and Dybvig (Journal of Political Economy, 1983, 91, 401 419) model, where banks would like to obtain insurance against shocks on returns on liquid assets through an interbank borrowing and lending program. We show that if investments in liquid assets and their realized returns are private information to individual banks, the first-best allocation is not incentive-compatible; we then characterize the second-best interbank contract.